There's a moment when Coolidge uses very specific Christian symbols to express the spiritual dimension of America's ideals and to represent the moral dimension of American policy. America has always been a predominantly Christian country, so if you're gonna use symbolism from one religion, it's a safe bet that people will get the references.

Early on, he establishes that being "American" and "enlightened" should include pursuing "a conscientious and religious life" (5.1); presumably, any religion will do. Peace will only be found through "the religious conviction of the brotherhood of man," and that "Parchment will fail, the sword will fail, it is only the spiritual nature of man that can be triumphant" (11.13-14). He makes religion a sort of tool that can replace other, more earthly ones, to bring people together peacefully.

At the end of the speech, Coolidge declares: "The legions which she sends forth are armed, not with the sword, but with the cross" (25.9). Now he's using a very specifically Christian symbol, the cross, which confirms which religion he's been talking about this whole time.

The cross, and the associated Christian references, serve as a way to make his ideas about morality more tangible. His audience was largely Christian, and they'd understand that imagery and the symbolism associated with it. They'd have understood that the cross replacing the sword was supposed to mean that America was aiming for a moral influence in the world, rather than one based on violence and aggression.